We return to the example begun in Preview Activity 8.1 to see how to derive the
ID: 2883651 • Letter: W
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We return to the example begun in Preview Activity 8.1 to see how to derive the formula for the amount of money in an account at a given time. We do this in a general setting. Suppose you invest P dollars (called the principal in an account paying r% interest compounded monthly. In the first month you will receive 12 ere r is in decimal form e.g., if we have 8% interest, we write 0.08 of the principal P in interest, so you earn 12 8.1. SEQUENCES 453 dollars in interest. Assume that you reinvest all interest. Then at the end of the first month your account will contain the original principal P plus the interest, or a total of P1 AP P 12 P (1+ 12 dollars. (a) Given that your principal is now P1 dollars, how much interest will you earn in the second month? If P2 is the total amount of money in your account at the end of the second month, explain why Po P1 1 12 12 (b) Find a formula for P3, the total amount of money in the account at the end of the third month in terms of the original investment P (c) There is a pattern to these calculations. Let Pn the total amount of money in the account at the end of the third month in terms of the original investment P. Find a formula for Pn.Explanation / Answer
a) we have first month total amount P1=P(1+r/12)
we have given P1 be the principal amount for second month and P2 be the total amount at the end of the second month
P2=P1+P1*(r/12)=P1(1+r/12)=P(1+r/12)2 since P1=P(1+r/12)
P2=P(1+r/12)2
b) P3=P2+P2*(r/12)=P2(1+r/12)=P(1+r/12)3 since P2=P(1+r/12)2
where P2 will be the principal amount at the end of the third month and P2*(r/12) is the interest at the end of the third month
P3=P(1+r/12)3 is the total amount of money in the account at the end of the third month,P is principal amount
c) Let Pn the total amount of money in the account at the end of the third month and
in first month P1=P(1+r/12)
in second month P2=P(1+r/12)2
we have P3=P(1+r/12)3 is the total amount of money in the account at the end of the third month,P is principal amount
substitute n=3 months
Pn=P(1+r/12)n
where Pn is the total amount at the end of the third month
n is total months,P is principal amount
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